= Solution
The <Jordan–Hölder theorem> applies to a <module> of finite length: if
$$
0=M_0\subsetneq M_1\subsetneq\cdots\subsetneq M_r=M
$$
is a <composition series of a module>, its successive quotients are <simple modules>. Any two such <composition series of a module> have the same length $r$, and their <composition factors> are pairwise isomorphic after a permutation, with multiplicities preserved. Every finite-dimensional <module> over $A$ has finite length, since strict inclusions increase vector-space <dimension>. A finite-dimensional <algebra over a field> does not by itself make every possible <module> finite-dimensional, so the finite-length hypothesis should be retained in the statement.
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